Sigma bond cylindrical symmetry

A sigma bond has cylindrical symmetry along the bond axis => rotation of groups joined by a single bond does not usually require a large amount of energy => free rotation. 

As predicted by elementary hybrid bonding theory, the multiple bonds of the chemist s Lewis-structure diagram are usually found to correspond to two distinct types of NBOs (1) sigma-type, having exact or approximate cylindrical symmetry about the bond axis (as discussed in Sections 3.2.5-3.2.7), and (2) pi-type, having a nodal mirror plane passing through the nuclei 44... 

A sigma bond is a molecular orbital that looks like an s-type atomic orbital when viewed down its axis and has cylindrical symmetry. A tt bond looks like a p-type of atomic orbital from the same... 

The a (sigma) bond has its maximum electron density along the line-of-centers joining the two atoms (below left). Viewed end-on, the u bond is cylindrically symmetrical about the line-of-centers. It is this symmetry, rather than its parentage, that defines the sigma bond, which can be formed from the overlap of two s-orbitals, from two p-orbitals arranged end-to-end, or from an s- and a p-orbital. They can also form when... 

An idealized single bond is a sigma bond—one that has cylindrical symmetry. In contrast, a p-orbital or pi-bond orbital has pi symmetry—one that is antisymmetric with respect to reflection in a plane passing through the atomic centers with which it is associated. In ethene, the pi-bonding orbital is symmetric with respect to reflection in a plane perpendicular to and bisecting the C-C bond, whereas the pi-star-anti bonding orbital is antisymmetric with respect to this operation. 

The bonding orbitals formed from Is-ls and 2s-2s interactions have another important feature in common they are all cylindrically symmetrical. In other words, if you look at the molecular orbital end-on, you can rotate it around the axis between the two atoms by any amount and it looks identical. It has the symmetry of a cigar, a carrot, or a baseball bat. Bonding orbitals with cylindrical symmetry like this are known as c (sigma) orbitals, and the bonds which result from putting two electrons into these orbitals are known as o bonds. The single bond in the H2 molecule is therefore a a bond. 

The carbon-carbon bond of ethane is a sigma bond with cylindrical symmetry, formed by two overlapping orbitals. (The carbon-hydrogen bonds are also sigma bonds. They are formed from overlapping carbon sp orbitals and hydrogen s orbitals.)... 

Because the 2p orbitals have a node at the atomic nucleus, the electron distribution of the VB wavefunction formed by their side-on overlap does not have cylindrical symmetry, but instead changes sign when rotated by 180° about the bond axis. Bonds that have this property are called pi (tt) bonds. Thus, the double bond in the oxygen molecule consists of a sigma bond and a pi bond, which are not equivalent. The existence of two different types of bonds in double bond formation is something that is not predicted by simple Lewis theory. A triple bond such as that in N2 is described within the VB approach as a sigma bond formed from the head-on overlap of 2p orbitals and two pi bonds formed by the overlap of the 2py and 2p orbitals on one N atom with their counterparts on the other atom (Figure 3.7). 

The two C atoms in ethane are ip -hybridized and they are joined by a sigma bond. As discussed in Chapter 4, sigma bonds have cylindrical symmetry, that is, the overlap of the sp orbitals is the same regardless of the rotation about the C—C bond. 

Sigma bond (Section 2.2) Any bond with cylindrical symmetry. 

Both MOs described in Fig. 1.10 have cylindrical symmetry about the internuclear axis. MOs of this type are called sigma (a) MOs. Since the inter-nuclear axis is usually defined as the z axis atomic orbital has a a-symmetry about this axis and thus it can be combined with other orbitals of the same symmetry (per orbitals), p and Py have however other symmetry. Rotation about the internuclear axis is not symmetric and there is a nodal plane containing this axis. These kind of orbitals are said to have 7r-symmetry. The linear combination of n atomic orbitals leads to bonding (ti ) and antibonding (tc ) MOs. Analogously atomic orbitals with symmetry 5, i.e. two nodal planes containing the internuclear axis, may be combined to give MOs with the same symmetry. That will be discussed separately in Sect. 1.3. The formation of dinuclear molecular orbitals with two different classes of symmetry is illustrated in Fig. 1.14. 

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